Name:________________________________
Name:________________________________ Date:_______ Period:_____
Transformations Ms. Anderle
Transformations:
A transformation is an operation that maps an original geometric figure, a preimage, onto a new figure called the image. A transformation can change the _______________, ________________, or __________________ of an object.
A specific type of transformation is called an ___________________. This transformation preserves _________________________.
Types of Transformations:
1. Translation
• A translation ____________________ an object a fixed distance in any direction.
• The preimage and the image (translated figure) have the same __________ and ____________. They also ____________________.
Example:
• Mapping Rule:
Examples:
1. T5,3 A(2,3) 2. T-2,1B(4,-2) 3. T0,-5K(-2,-4)
4. T-3,4 G(x,y) 5. T5,4J(2k, k) 6. T-1,2T(2x, y)
7. If the image after T2,4 is (3,4) then what is the preimage?
8. If the image after T-2,-3 is (-2, -1) then what is the preimage?
9. If translation T maps point P(2,3) onto P’(-3,4), then find translation T.
10. If translation T maps point L(5,14) onto L’(-5,-2), then find translation T.
2. Line Reflections
• A line reflection is a transformation where an image is ________________ over a given line.
• Each point in the reflection is the ________________ from the line of reflection as the points in the original figure.
Example:
Line Reflection Mapping Rules
|Reflections in the x-axis | |
|Reflections in the y-axis | |
|Reflections in the y = x line | |
|Reflection in the y = -x line | |
|Reflection in the line y = k | |
|(where k is a constant) | |
|Reflection in the line x = k | |
|(where k is a constant) | |
1. Find the reflection in the y = -x line of ΔPQR with the coordinates P(2,2), Q(5,3), and R(3,-1).
2. Find the reflection of the figure whose coordinates are A(-1,3), B(4,6), C(7,2) across the line represented by x = 2.
3. Find the reflection of the figure whose coordinates are A(-3,-7), B(3,-4), and C(9,-6) across the line represented by y = -3.
4. State the coordinates of the point (-2,5) under the reflection
a. in the line y = x
b. in the line y = 4
c. in the line x = -2
5. Find the image of the points below in reflected in the line y = -2
a. (1, 4)
b. (-3, 3)
c. (4, -5)
3. Point Reflection and Symmetry
• If a figure is reflected in or through point P, then P is _______________ of the line segement joining each point to its corresponding image.
• Point Reflection Mapping Rules:
-Over the Origin
-Over a Given Point:
A figure is said to have point symmetry if the figure coincides with itself when reflected through a point or when rotated 180° about a point. The point that is the center of reflection or rotation is called the point of symmetry.
| |F, G, |
|Shapes without point symmetry | |
| |Z, S, N, |
|Shapes with point symmetry | |
Examples:
1. What letter has both point symmetry and line symmetry?
a. A b. H c. E d. S
2. What is the image of (k, 2k) after a reflection through the origin.
3. Find the image of (-4, 2) under the reflection in the point (2, 2).
4. Find the image of (-2, -6) under a point reflection in the origin.
5. Find the image of (3, 2) under the reflection in the point (-1, -1)
4. Rotations:
• A rotation is a transformation in which a figure is __________________ point called the __________________.
• In the coordinate plane, this point is typically the origin.
• Rotations that are counterclockwise are rotations of ____________________ degree measure.
• Rotations that are clockwise are of _________________ measure.
• All rotations are assumed to be clockwise about the origin unless otherwise stated
Example:
[pic]
Rotation Mapping Rules
|Rotation of 90° and -270° | |
|Rotation of 180° and -180° | |
|Rotation of 270° and -90° | |
|Rotation of 360° and -360° | |
Examples:
1. What is the image of (-2, 3) under a rotation of:
a) 90° b) 180° c) 270° d) -90° e) -270°
2. Name the coordinates of each point under a rotation of 90°
a) (5,1) b) (-3,3) c) (8, -2) d) (0,5)
3. Name the coordinates of each point under a rotation of 270°
a) (-3,2) b) (-2,-2) c) (4,4) d) (7,14)
5. Dilations
• A dilation is a transformation in which ___________ of a figure is changed and the figure is moved.
• Dilations are ______________________________________________.
Mapping Rule:
Model Problems:
1. D3(x, y) = (3x, 3y) 2. D1/2(2, 4) = (1, 2)
Note:
-If k = 1, then the image is identical to the original.
-If k = -1, then the image is congruent and is the same as a point reflection.
-If |k| > 1, then the image is similar and larger than the original.
-If |k| < 1, then the image is similar and smaller than the original.
Examples:
1) What are the coordinates of point (2, -4) under the dilation D-2?
2) What are the coordinates of point (9, -6) under the dilation D1/3?
3) What are the coordinates of point (12, -4) under the dilation D5?
4) If the dilation Dk(2, -4) = (-1, 2), the scale factor k is equal to:
5) Find the image of (6, -9) under the dilation D2/3.
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